The Formation of a Disk Galaxy Within a Growing Dark Halo

Home , Dark matter halo, Galactic halo, Thick disk

A&A 399, 961–982 (2003) Astronomy DOI: 10.1051/0004-6361:20021842 & c ESO 2003 Astrophysics

The formation of a disk galaxy within a growing dark halo

M. Samland1 andO.E.Gerhard1

Astronomisches Institut der Universit¨at Basel, Venusstrasse 7, 4102 Binningen, Switzerland

Received 3 July 2002 / Accepted 29 November 2002

Abstract. We present a dynamical model for the formation and evolution of a massive disk galaxy, within a growing dark halo whose mass evolves according to cosmological simulations of structure formation. The galactic evolution is simulated with a new three-dimensional chemo-dynamical code, including dark matter, stars and a multi-phase ISM. The simulations start at redshift z = 4.85 with a small dark halo in a ΛCDM universe and we follow the evolution until the present epoch. The energy release by massive stars and supernovae prevents a rapid collapse of the baryonic matter and delays the maximum star formation until redshift z ≈ 1. The metal enrichment history in this model is broadly consistent with the evolution of [Zn/H] in damped Lyα systems. The galaxy forms radially from inside-out and vertically from halo to disk. As a function of metallicity, we have described a sequence of populations, reminiscent of the extreme halo, inner halo, metal-poor thick disk, thick disk, thin disk and inner bulge in the Milky Way. The first galactic component that forms is the halo, followed by the bulge, the disk-halo transition region, and the disk. At redshift z ≈ 1, a bar begins to form which later turns into a triaxial bulge. Despite the still idealized model, the final galaxy resembles present-day disk galaxies in many aspects. The bulge in the model consists of at least two stellar subpopulations, an early collapse population and a population that formed later in the bar. The initial metallicity gradients in the disk are later smoothed out by large scale gas motions induced by the bar. There is a pronounced deficiency of low-metallicity disk stars due to pre-enrichment of the disk ISM with metal-rich gas from the bulge and inner disk (“G-dwarf problem”). The mean rotation and the distribution of orbital eccentricities for all stars as a function of metallicity are not very different from those observed in the solar neighbourhood, showing that early homogeneous collapse models are oversimplified. The approach presented here provides a detailed description of the formation and evolution of an isolated disk galaxy in a ΛCDM universe, yielding new information about the kinematical and chemical history of the stars and the interstellar medium, but also about the evolution of the luminosity, the colours and the morphology of disk galaxies with redshift.

Key words. galaxies: formation – galaxies: evolution – galaxies: stellar content – galaxies: structure – galaxies: kinematics and dynamics – galaxies: ISM

1. Introduction disagreement with the flat dark matter density distributions inferred from observations in the centres of galaxies (Salucci & During the last decade, significant progress has been made in Burkert 2000; Blais-Ouellette et al. 2001; Borriello & Salucci understanding cosmic structure formation and galactic evolu- 2001; de Blok et al. 2001). Secondly, the specific angular motion. With high-resolution cosmological simulations, the for- ff menta and scale lengths of the disk galaxies in the cosmological mation of dark halos in di erent cosmologies has been studied simulations are too small compared to real galaxies (Navarro in detail (e.g. Navarro et al. 1996; Moore et al. 1998; Col´ın et al. & Steinmetz 2000b). This may be partially a numerical prob- 2000; Yoshida et al. 2000; Avila-Reese et al. 2001; Jenkins lem of SPH simulations, but is more likely due to overly ef- et al. 2001; Klypin et al. 2001) to determine the large-scale ficient cooling and strong angular momentum transport from mass distribution, the halo merging histories, the structural pa- the baryons to the dark halos during merger events (Navarro & rameters of the dark halos, and finally the formation of galaxies Benz 1991). Thacker & Couchman (2001) and also Sommer- inside these dark halos (Steinmetz & Müller 1995; Kay et al. Larsen et al. (1999) showed the angular momentum problem 2000; Navarro & Steinmetz 2000b; Mosconi et al. 2001; Pearce in CDM simulations may be resolved by the inclusion of feed- et al. 2001). back processes. Thirdly, the cold dark matter (CDM) simula- In many respects, these simulations are in agreement with tions predict a large number of satellite galaxies in the neigh- ffi observations, but there remain a number of di culties. Firstly, bourhood of giant galaxies like the Milky Way Galaxy, but this the universal cuspy halo density profiles (Navarro et al. 1997; is not observed (Moore et al. 1999). Moore et al. 1999; Fukushige & Makino 2001), while in agree- The cosmological simulations are used to describe the ment with observations of galaxy clusters, are in apparent large-scale evolution of the dark matter distribution from Send offprint requests to: M. Samland, the primordial density fluctuations to the time when the e-mail: [email protected] dark halos form. It is inevitable in such simulations that

Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20021842 962 M. Samland and O. E. Gerhard: Formation of a disk galaxy structures on galactic and sub-galactic scales are not well- discussed in Sect. 5. A summary and a short outlook conclude resolved. Furthermore, it is difficult to incorporate the detailed the paper (Sect. 6). physics of the baryonic component, that is of the multi-phase interstellar medium (ISM), and the feedback processes between 2. Observations of the galaxy formation process stars and ISM. These are serious drawbacks, because feedback from stars can prevent a proto-galactic cloud from rapid col- Galaxy evolution models can give a simplified description of lapse and it can trigger large scale gas motions (Samland et al. the evolution of real galaxies and can make predictions that can 1997). Both alter the galactic formation process significantly. be tested by observations. The high redshift observations show Complementary ways to circumvent at least some of that the first proto-galactic structures form at redshift z > 3 these problems are (i) to combine the large-scale simulations (e.g. Clements & Couch 1996; Dey et al. 1998; Steidel et al. with semi-analytical galaxy models (Guiderdoni et al. 1998; 1999; Manning et al. 2000). Surveys of high redshift galax- Somerville & Primack 1999; Boissier & Prantzos 2000; Cole ies reveal a wide range of galactic morphologies with con- et al. 2000; Kauffmann & Haehnelt 2000), or (ii) to simulate siderable substructure and clumpiness (Pentericci et al. 2001). the formation of only single galaxies or small galactic groups At redshift z ≈ 1−2, the formation of galactic components using either cosmological initial conditions or a simplified, (spheroids, halos, bulges and disks) seems to take place (Lilly but cosmologically motivated model of the dark matter back- et al. 1998; Kajisawa & Yamada 2001). However, the spatial ground (Navarro & Steinmetz 1997; Berczik 1999; Hultman resolution of these high and intermediate redshift surveys (e.g. & Pharasyn 1999; Sommer-Larsen et al. 1999; Bekki & Chiba Abraham et al. 1999; van den Bergh et al. 2000; Ellis et al. 2001; Thacker & Couchman 2001; Williams & Nelson 2001). 2001; Menanteau et al. 2001) is still too low to measure more While the semi-analytical models are advantageous for inves- than global galactic properties (e.g. asymmetry and concentra- tigating the global properties of galaxy samples, the small- tion parameters). There are indications that most of the mas- scale dynamical models provide information about the detailed sive galaxies form before or around z = 1 (Brinchmann & Ellis structure of galaxies, the kinematics of the stellar populations 2000), and especially that the giant ellipticals form at redshifts and the ISM, and the star formation histories. The dynam- z > 1 (Best et al. 1998), (but see van Dokkum et al. 1999), but ical models are called chemo-dynamical (Theis et al. 1992; not before z = 2.4 (Pentericci et al. 2001). There is a general Samland et al. 1997; Bekki & Shioya 1998; Berczik 1999; trend of early type galaxies being older and having higher peak Williams & Nelson 2001), if they include different stellar popu- star formation rates than late type galaxies (Sandage 1986). lations, a multi-phase ISM, and an interaction network that de- These general star formation histories seem to be overlapped scribes the mass, momentum and energy transfer between these with star formation bursts, triggered by infalling material or in- components. teractions with other galaxies. The aim of the present work is to investigate how a large Since we can observe galactic sub-structures down to the disk galaxy forms inside a growing dark halo in a realistic stellar scale only in our Galaxy, the Milky Way observations are ΛCDM cosmogony. We present the results of a new three- a cornerstone for the understanding of the formation and evo- dimensional model, including dark matter, stars, and a multi- lution of all spiral galaxies. From studies of Galactic stars we phase ISM. The mass infall at the boundaries of the simulated know that the halo is old and that the disk and bulge contain a volume is taken from cosmological simulations (see VIRGO- mixture of old and young stars. This raises the question in what GIF-project; Kauffmann et al. 1999). We use a total dark matter sequence the galactic components halo, bulge and disk formed, 12 11 mass of 1.5 × 10 M, a total baryonic mass of 3 × 10 M, and what was the star formation history. Determinations of a spin parameter λ = 0.05, and an angular momentum pro- the star formation history of the Milky Way and other nearby file similar to the universal profile found by Bullock et al. galaxies were attempted by Bell et al. (2000); Hernandez et al. (2001). Different from Bekki & Shioya (1998); Berczik (1999) (2000) and Rocha-Pinto et al. (2000), based on star counts, and Williams & Nelson (2001), we use an Eulerian grid-code broad band colours, or chemical compositions of stars. for the ISM, which we believe is better suited to describe the The chemical and kinematical data from galactic halo stars multi-phase structure of the ISM and the feedback processes, have been one of the starting points for different scenarios for especially in low-density regions. The final galaxy model pro- the formation and evolution of galaxies. Eggen et al. (1962) vides densities, velocities, velocity dispersions, chemical abun- were the first to propose a monolithic collapse galactic forma- dances, and ages for 614 500 stellar particles, and temperatures, tion model in which the Milky Way forms rapidly out of a col- chemical abundances, densities, velocities, and pressures of the lapsing proto-galactic cloud. This was based on data for galac- ISM. The goal of the project is a self-consistent model for the tic halo stars which are now known to have been plagued by formation and evolution of large disk galaxies, which can be selection effects (Chiba & Beers 2000). Later, based on ob- used to understand observations of young galaxies as well as servations of halo globular clusters, Searle & Zinn (1978) pro- data on stellar populations in the Milky Way. posed a new model in which the Milky Way forms out of merg- The paper is organized as follows. Section 2 summarizes ing lumps and fragments which all have their individual star some observational material relating to the formation and evo- formation histories. The contrasting nature of these two models lution of galaxies. Section 3 describes our multi-phase ISM and the proposed observable differences (e.g. existence or non- model and the underlying dark matter halo model. In Sect. 4 existence of metallicity gradients, the expected age spread for we outline the formation history of the model disk galaxy. The halo objects) seemed to offer a simple way to deduce the for- properties of the galactic halo, bulge and disk components are mation history of disk galaxies from Milky Way observations. M. Samland and O. E. Gerhard: Formation of a disk galaxy 963

However, new data especially from proper motion catalogues, lead to the conclusion that at least the halo of the Milky Way formed neither by a monolithic collapse nor by a pure merging process (Sandage 1990; Unavane et al. 1996; Chiba & Beers 2000). In the modern hybrid scenarios, a part of the Galactic No.35 No.22 No.47 20 halo formed during a dissipative process (collapse) and another part shows signatures of merging events. For example, Chiba & Beers (2000) proposed a scenario in which “the outer halo is made up from merging and/or accretion of sub-galactic ob- 15 ] jects, such as dwarf-type satellite galaxies, whereas the inner o part of the halo has undergone a dissipative contraction on rel- M 11 atively short time scales”. This is consistent with the findings 10 of Helmi et al. (1999), who found that 10% of the metal-poor stars in the outer halo of the Milky Way are aligned in a sin- mass [10 gle coherent structure, which they identify as the remnant of a 5 disrupted dwarf galaxy. These hybrid scenarios are, in the general outline, also consistent with the formation of galaxies in a hierarchical universe (Chiba & Beers 2000). 0 0 2 4 6 8 10 12 14 3. The model time [Gyr] The formation and evolution of a galaxy is influenced by the Fig. 1. The dots shows the average mass from 96 different halo merg- 12 cosmological initial conditions and the environment, but also ing trees, which all result in a 1.8 × 10 M halo. The smooth line by internal feedback processes. Heating by supernovae (SNe), shows the approximation which is used in our numerical model. The dissipation, radiative cooling, as well as in- and outflows in- most extreme halo merging trees and that which shows the smallest deviations from the average are shown in the smaller upper panels. fluence the evolution both locally and globally. Our new three- Data from the VIRGO-GIF-project (Kauffmann et al. 1999). dimensional galactic model takes into account the dynamics of stars and the different phases of the ISM, as well as the processes (“chemistry”) which connect the ISM and the stars. a galaxy for a large number of different halo formation histo- These baryonic components are embedded in a collision-free ries. We choose a ΛCDM universe with Λ=0.7, Ω=0.3, dark matter halo taken from simulations of cosmological struc- a Hubble constant h0 = 0.7 and a baryonic-to-dark matter ra- ture formation. The description of the ISM is based on that used tio of 1:5. From the cosmological N-body simulations of the in the two-dimensional chemo-dynamical models of Samland VIRGO-GIF-project (Kauffmann et al. 1999), we extracted the 12 & Hensler (1996); Samland et al. (1997). The following sub- merging trees of 96 halos, all with a final mass of 1.8×10 M. sections contain a brief outline of our model. Figure 1 shows the average halo mass growth, which is in agreement with the universal mass accretion histories found by van den Bosch (2002) and Wechsler et al. (2002). The individ- 3.1. The dark halo ual merging histories can be quite different; two extreme cases Cosmological simulations (e.g. Navarro et al. 1995; Moore with early and late mass growth are shown in the small panels et al. 1999; Pearce et al. 1999) show that in a hierarchical uni- at the top left and right of Fig. 1. verse galaxies form out of a few large fragments with an ad- We assume that the growing dark halo has the universal ditional slow accretion of mass. Thus each galaxy has its own density profile as proposed by Navarro et al. (1997), and that formation history and characteristic shape, depending on the its mass is accreted in spherical shells. Given the current halo masses, angular momenta and merging histories of the frag- mass and the evolution of the critical density, we can determine ments. Yet the formation of the CDM halos can be described the outer halo radius r200 (see Mo et al. 1998). A free parameter by some simple “universal” formulae. Wechsler et al. (2002) that remains is the mass concentration c, which in the present and van den Bosch (2002) calculated the mass accretion his- simulation is assumed to be c = 9.2. At redshift z = 0, this leads −1 tory, Bullock et al. (2001) the angular momentum distribu- to a circular velocity of vcirc = 179 km s at the r200 = 250 kpc tion, Navarro et al. (1997) and others the density profiles and radius. Inside a galactocentric radius of 8.5 kpc, the final mass 10 Del Popolo (2001) and Wechsler et al. (2002) the evolution of of dark matter is 4 × 10 M, comparable to the upper limit of 10 the concentration parameter. the dark matter content inside the solar circle (5.2 × 10 M; Navarro & Steinmetz 2000a). Treating the growing dark matter halo as a spherical back- 3.1.1. The dark matter halo formation history ground potential that does not respond to the baryonic com- In the hierarchical formation scenario, a dark halo may form ponent minimizes gravitational torques during the collapse early and rapidly or grow slowly with time. For the model de- and thereby circumvents the well-known angular momentum scribed in this paper we take an average halo formation his- problem (Navarro & Steinmetz 2000b). The accretion in this tory, because at present we cannot simulate the evolution of model is smoother as compared to hierarchical models, where a 964 M. Samland and O. E. Gerhard: Formation of a disk galaxy fraction of the mass accretion takes place through merging dark ] matter fragments, unless the gas in these hierarchical fragments 2 1000 /pc is dispersed by early star formation and feedback. o 100 3.1.2. The angular momentum of the dark matter 10 During the hierarchical clustering, tidal torques induce shear N flows which lead to rotating proto-galactic systems. -body 1 simulations show that, independent of the initial perturbation surface density [M spectrum and the cosmological model, the spin parameter of 0 10 20 30 40 50 . − − . the dark halos is in the range λ = L|E|0 5G 1 M 2 5 = 0.02−0.11 r [kpc] with an average value of λ = 0.04−0.05 (Barnes & Efstathiou Fig. 2. Surface density of the galactic disk if the specific angular mo- 1987; Steinmetz & Bartelmann 1995; Cole & Lacey 1996; mentum is conserved during the proto-galactic collapse in our NFW van den Bosch 1998; Gardner 2001). halo model. The cylindrical rotation leads to an exponential disk in the Bullock et al. (2001) studied the angular momentum pro- inner 20 kpc (dashed curve), while the spherical rotation produces an files of dark halos in a ΛCDM cosmology. They found that the exponential disk with an additional mass at the centre (full line). The spatial distribution of the angular momentum in most of the thin dash-dotted line shows an exponential disk with a scale length of halos show a certain degree of cylindrical symmetry, but that 8 kpc. a spherically symmetric angular momentum distribution with apower-lawj(r) ∝ r1.1±0.3 is also a good approximation. We consider the following two rotation fields for the halo: star formed and the evolution of the galactic gravitational potential. We use a particle-mesh method (Hockney & Eastwood x2 + y2 | | vhalo = v ≡ v r 1988), in which the passing of the stellar particles to the mesh rot 0 0 (1) 2 2 2 | | + and also the interpolation of the gravitational forces at the parti- x + y + z + d0 d d0 cle positions is done with a cloud-in-cell method. This method x2 + y2 |r| is fast and second order accurate, but it is limited by the spatial vhalo = v ≡ v rot 0 0 | | + (2) resolution of the underlying grid, which at present is 370 pc. x2 + y2 + d r d0 0 Each stellar particle represents an ensemble of stars with v0 and d0 are constants which determine the total amount of an- a stellar mass function. In the present model, we assume a gular momentum and the rotation curve near the rotation axis, Salpeter initial mass function (IMF) with lower and upper mass respectively. limits of 0.8 M and 50 M, and with an additional lock-up Assuming that initially the mass and velocity distributions mass fraction of 60%. The lock-up mass equals the mass of of the baryonic matter coincide with those of the dark mat- all stars below 0.8 M. During a simulation the star forma- ter, and that baryonic mass and specific angular momentum is tion rate is calculated on the spatial grid, and in timesteps of conserved during the collapse to a disk (Mestel 1963; Fall & 10 Myr representative samples of 500 new stellar particles are Efstathiou 1980), we can estimate the disk surface density pro- created. This yields 614 500 particles at the end of the simula- files corresponding to these rotation fields. Figure 2 shows the tion. Depending on the star formation rate a single particle rep- results, calculated for collapse in the pure NFW potential. As resents 104 to 107 stars. The initial chemical composition and already pointed out by Fall & Efstathiou (1980), disks with ex- velocity of a stellar particle is derived from the chemical com- ponential surface density profiles are found when the halo has position, velocity and velocity dispersion of the parent molec- a rising rotation curve at small radii and a nearly constant rota- ular cloud medium. tion velocity in the outer parts. In the present case, the cylindri- We assume that the evolution of the stars is determined only cal halo rotation model (Eq. (2)) also produces an exponen- by their initial mass and metallicity, neglecting effects of stel- tial disk, at least in the inner 20 kpc. The spherical rotation lar rotation or star-star interactions. From the stellar evolution model (Eq. (1)) instead leads to higher surface densities near models of Maeder & Meynet (1989); Schaller et al. (1992) and the galactic centre which favours the formation of a bulge. In Schaerer et al. (1993) we get the lifetimes, energy releases and the following we use the spherical rotation model with a spin mass losses on the main sequence. We can subdivide stars into parameter λ = 0.05, to fix the initial angular momentum distri- three main classes. bution of the baryonic matter. First, the low mass stars which stay on the main sequence during the whole galactic evolution. These stars become notice- able only by their gravitational forces and because they lock-up 3.2. The stars a major fraction of the galactic mass. The total lock-up mass The stellar motions have a considerable influence on the galac- has influence on the galactic gas content and therefore on the tic evolution, since the time-delayed stellar mass loss and en- sequence of stellar generations. It is important in smaller galax- ergy release connect different regions and also different galac- ies where SN-driven winds reduce the mass in gas, but not the tic evolution phases. The approximately 1011 stars in a large mass locked up in low mass stars. galaxy form a collision-less system. Each star moves on its own The second class of stars are intermediate in mass. When orbit, which is determined by the conditions under which the they leave the main sequence, they loose a substantial fraction M. Samland and O. E. Gerhard: Formation of a disk galaxy 965 of the initial mass (asymptotic giant branch and planetary neb- and energies (dimensional splitting). Using the van Leer ad- ula phase); however, the energy release of these stars is small vection scheme (van Leer 1977) combined with the consistent compared to other energy sources. advection method (Norman et al. 1980) and the Strang split- The third class of stars are the massive stars, typically with ting (Strang 1968), the advection is second order accurate. In masses in excess of 8–10 M. These stars are a major source of the Strang splitting the three-dimensional transport is split into both energy and mass return in a galaxy. The UV-radiation and a sequence of five one-dimensional transport steps: the kinetic energy release during the final explosion (SN type ∆t ∆t ∆t ∆t II) ionize and accelerate the ISM. This can trigger large-scale ∆tA = Ax Ay (∆tAz) Ay Ax (4) gas flows and star formation which both influence the galactic 2 2 2 2 evolution. where ∆t is the time step length, Ax, Ay, Az are the differ- A rare but important galactic event is the explosion of SNe ential operators which describe the transport in x, y, z direc- of type Ia. The most probable progenitor candidates are close tion, and A is the operator for the three-dimensional problem. binary stars consisting of a white dwarf and a main sequence Approximately the same accuracy can be achieved by a simpler star. We include this process, since it is a major source of iron splitting scheme which uses an alternating sequence of only peak elements (Nomoto et al. 1984). three one-dimensional transport steps: (∆tA)(∆tA) = (∆tAx)(∆tAy)(∆tAz) 3.3. The Interstellar Medium (∆tAz)(∆tAy)(∆tAx) . (5) The ISM in galaxies exists at different densities and temperatures, and it shows phase transitions on short timescales com- For the ∼106 transport steps in our galactic evolution simula- pared to the age of a galaxy. In the present galactic model tions, both splitting schemes lead to identical results. However, we use a simplified ISM description based on the three-phase the second method is faster by a factor of 1.6. model of McKee & Ostriker (1977). The three phases are a cold, a warm and a hot phase, with typical temperatures of 3.4. The interaction network 80 K, 8000 K and 106 K, respectively. The cold phase is found in the dense cores of molecular clouds. These cores are em- The interactions between stars and ISM are very important for bedded in warm neutral or ionized gas envelopes. Outside the the galactic evolution. They determine the evolution timescale clouds the space is filled with hot, dilute gas. The three-phase of a galaxy to a large extent. In the following subsections we model is a simplification, but it is realistic enough to use it in give a short overview about the processes which we take into a global galactic model. A critical discussion of the weak and account, and discuss the influence of the free parameters in the strong points of the three-phase model can be found in McKee parametrization of these processes. (1990). Essentially, the temperatures and densities of the ISM 3.4.1. Star formation phases determine most of the physical processes in the ISM. However, the geometry of the clouds is also important for pro- Star formation is obviously one of the most important processes cesses that take place in the phase transition regions (e.g. evap- during the galactic evolution, but it is one of the least under- oration and condensation) or that depend on the sizes of the stood. Early on, Schmidt (1959) found that the star formation clouds explicitly (e.g. cloud-cloud collisions and star forma- rate in the solar neighbourhood varies with the square of the tion). In our model, we assume spherical clouds which follow surface gas density Σgas of the ISM. More recently, Kennicutt the mass-radius relation of Elmegreen (1989) (1998) found a star formation rate for disk and starburst galax- Σ1.4±0.15 ies which is proportional to gas , with a sharp decline M M −2 = P . below a critical surface density threshold (≈3–13 M pc ). 2 190 2 4 (3) R pc Remarkably, this simple law describes star formation in galax- / ies for which the average gas consumption time can vary from P = Pe k is the pressure in the ambient ISM. The mean 4 104 K/cm3 3 × 108 yr (starbursts) to 2 × 109 yr (normal disks). 5 cloud mass is 10 M; this is derived from the cloud mass spec- Motivated by these findings we use the following star for- trum of Dame et al. (1986) for star forming giant molecular mation law: 4 6 clouds in the mass range of 10 M to 10 M. ρcld ρcld In the present dynamical description, we use only a two- ρ˙sf = = · (6) t c t phase model for the ISM, containing hot gas with embed- sf sf dyn ded (cold+warm) clouds. The motions of both components Here ρcld is the average density of the cloudy medium, tsf the are described by the time-dependent hydrodynamical equa- star formation timescale, and tdyn the dynamical timescale for tions (mass, momentum and energy conservation). Similar as in the average star forming cloud mass. For the cloud model de- the two-dimensional chemo-dynamical models (Samland et al. scribed in Sect. 3.3 the dynamical timescale of the average star 1997) we use a fractional step method to split the problem forming cloud is into source, sink and transport steps. The problem of the three- π dimensional transport is broken down into a number of one- = 3 = . · −3/8. tdyn 16 2 P4 (7) dimensional problems for the densities, metallicities, momenta 32Gρˆcld 966 M. Samland and O. E. Gerhard: Formation of a disk galaxy

Molecular clouds do not collapse and form stars on a single dynamical timescale, because magnetic fields or turbulent mo- 2 tions stabilize the clouds. To account for these processes, we include the factor c . If we take into account that a part of sf starburst the clouds is only confined by pressure and that the timescales galaxies for ambipolar diffusion and dissipation of turbulent energy are 1 much longer than the dynamical timescale, we expect csf 1. csf can be estimated from the observed number of OB stars in 8 the solar neighbourhood (Reed 2001). Assuming that the stel- ]

-2 9 lar mass function follows a Salpeter law with lower and upper 0 12131110

mass limits of 0.1 M and 50 M, we derive a local star for- kpc −5 −3 −1 -1 mation rate of 1.8 × 10 M pc Myr . From the local ISM 76 yr 5 −3 −3 o density and pressure (ρ ≈ 0.035 M pc ≈ 1cm , P4 ≈ 1), [M we find csf = 120, which means that the on average the star -1 SFR formation timescale exceeds the dynamical timescale by factor Σ 76 of 120. The final star formation law can then be written as log 98 4 11105 12 ρ = η ρ 3/8 13 ˙sf sf cldP4 (8) -2 2 with a star formation (gas depletion) timescale tsf ≈ 1/ηsf = 4 disk galaxies 1.9 Gyr in the local Galactic disk. The star formation rate is proportional to ρ1.375 if the cloudy medium is in pressure equi- cld -3 3 librium (P4 ∝ ρcld). This is the quiescent star formation mode 3 in the model. During the collapse or if large fragments merge, 2 the pressure can increase and the star formation can be more efficient. 0 1 2 3 4 log Σ [M pc-2] Figure 3 shows the resulting star formation rate as a func- gas o tion of the surface density of gas, in the forming disk galaxy model described in Sect. 4. Both the average star formation Fig. 3. Star formation rate in the forming disk galaxy model described in Sect. 4, versus the projected surface density of the cloudy medium rates and the star formation rates in the strongest star formation in a 20 kpc sphere. Open numbered circles are for average star for- regions are slightly higher than, but consistent with Kennicutt’s mation rates and surface densities within 20 kpc; the number denotes 1998 data and mean relation, over a range of surface densities the galactic age in Gyr. The filled numbered circles are for the most extending from normal disk galaxies into the starburst galaxy prominent star formation region at the same galactic ages. The black region. Had we calibrated the factor ηsf on Kennicutt’s diagram, dots and black squares denote the data from Kennicutt (1998) for disk the resulting value would have been about a factor of 2 smaller galaxies and starburst galaxies. The dashed line is Kennicutt’s mean than that based on the solar neighbourhood OB stars. The typ- relation. ical scatter in Kennicutt’s plot corresponds to a factor of 2.2 dispersion in ηsf.

3.4.2. Massive stars and supernovae type II (i) We assume that the mass return of a massive star takes place instantaneously at the end of its evolution, when the SNe Massive stars and SNe influence the galactic evolution in sev- explodes. This is a good approximation, because the time be- eral ways. They heat the ISM, produce shells and filaments and tween the stellar mass loss by a wind and the ejection of the SN stir up the cloudy medium. In addition, they are the most impor- shell is short compared to galactic timescales. tant source of heavy elements and they determine the chemical evolution of a galaxy. The lifetimes of the massive stars range (ii) We assume that the UV photons emitted by the massive from 3 to 30 Myr before they finally explode as SNe II. Only stars are absorbed and reradiated by the dense cloud medium. a stellar remnant of ≈2 M remains. The rest of the mass is Since the cooling time of this gas is short compared to the life- blown away by stellar winds or expelled during the explosion. time of the massive stars (the typical timescale which we re- On average a massive star on the main sequence emits about solve in our models), we do not simulate this heating and cool- 6 × 1051 erg in photons (Schaerer & Vacca 1998) which ionize ing process, but assume that the clouds are in thermal equilib- the surrounding ISM and regulate the star formation. During rium. the SN explosion the star releases another 1051 erg, creating a (iii) The SN explosion energy (1051 erg) is released in a fast expanding bubble. This influences the star formation pro- short time and produces a bubble of hot gas and an expand- cess, since it decreases the gas density in the surroundings of ing shell. The shell interacts with the clouds and by this in- the SN and it can trigger star formation in other regions. creases the velocity dispersion σcld of the cloudy medium. In the model we can use only a simple description of these The SN energy is released locally, and we assume that ηicm = processes. 95% of the SN energy heats the intercloud gas, and that the M. Samland and O. E. Gerhard: Formation of a disk galaxy 967 remaining ηcld = 5% goes into the velocity dispersion of the gas (van den Hoek & Groenewegen 1997) and, in total, they cloudy medium (McKee & Ostriker 1977). Thus return twice as much mass to the ISM as the massive stars, however with a much lower energy and a significant time de- ˙ = η · × 51 ≈ η · ρ · , Eicm icm n˙sn 10 erg icm ˙sf Esn (9) lay. In the model we neglect the radiation of the intermediate 51 E˙cld = ηcld · n˙sn × 10 erg ≈ ηcld · ρ˙sf · Esn. (10) mass stars, because they do not heat the ISM efficiently, but include the mass return and the enrichment in terms of a third In the second half of these equations we have related the fiducial element. heating rates to the star formation rate; here Esn = 2.7 × 105 pc2 Myr−2 is the specific SN energy (per mass of stars formed for one supernova II to occur). The parameters ηicm, 3.4.5. Radiative cooling of the hot gas η and E are not free, but are fixed in a narrow range by cld sn The heating of the ISM by SNe is mainly balanced by radiative observations and supernova models. energy losses. This cooling process determines the temperature (iv) SNe are the most important sources of heavy elements. and pressure of the hot ISM and by this has influence on the dy- The lack of self-consistent explosion models of massive stars namics of the ISM. We use the metallicity and temperature de- causes uncertainties in chemical yields by factors of two and pendent cooling functions Λ(T , Z ) of Dalgarno & McCray more. However, the detailed chemical composition is not im- icm icm (1972) and Sutherland & Dopita (1993). These cooling func- portant for the galactic evolution. We therefore use a fiducial tions provide lower limits to the real energy loss, because they chemical element which traces the fraction of heavy elements are calculated for a homogeneous gas in thermal equilibrium. produced in the hydrostatic burning phases and during the SN In a real ISM with density fluctuations, the cooling rate can be explosion of massive stars. With the yield tables of Woosley enhanced by a factor η = 2.3−10 (McKee & Ostriker 1977), & Weaver (1995) and Thielemann et al. (1996), or the yield cool so that approximations of Samland (1998) we can convert the abundances of the fiducial SN II element into real chemical abun- E˙ = −η · ρ2 Λ(T , Z ). (11) dances. This method has the advantage that it is not necessary icm cool icm icm icm redo the dynamical simulation when new yield tables become We use an enhancement factor of ηcool = 5todescribethera- available. diative energy loss in the simulations. The radiative energy loss E˙icm scales with the square of the gas density ρicm because the 3.4.3. Supernovae type Ia ions are excited by collisions. The progenitors of type Ia SN are believed to be close binary stars consisting of an intermediate mass star and a white 3.4.6. Dissipation in the cloudy medium dwarf. In the present model only a small fraction of the 1.5 M– The cloudy medium is described as a hydrodynamic fluid char- 8.0 M stars explode as SNe Ia. This fraction is determined by acterized by its density, momentum, and kinetic energy. The the number ratio of SNe type Ia to type II. We take a ratio of individual clouds are assumed to be at a thermal equilibrium 1:8.5 (Samland 1998) which is consistent with the observed temperature. The kinetic energy density of the cloudy medium SN rates in galaxies (Tammann et al. 1994) and can explain has source and sink terms from the heating by SNe and dissi- the observed iron abundances and the Galactic evolution of the pation by cloud-cloud collisions. The description of the cloud- [α/Fe] ratio. In our model both the energy released by a SN Ia 51 cloud collisions is based on the inelastic cloud collision model (10 erg) and the mass of the white dwarf progenitor (0.6 M) of Larson (1969). There, the energy loss of the cloudy medium are given to the hot ISM. Because the mass return of SN Ia is depends on the effective cross section Σ=c ·πR2, which dif- small compared to other stars and the total energy released is coll fers from the geometrical cross section by the factor c .For an order of magnitude smaller than that from SNe type II, the coll a medium with density ρ and velocity dispersion σ con- SNe Ia do not influence the dynamical evolution significantly. cld cld sisting of clouds of mass M and radius R, the kinetic energy However, SNe type Ia are important for the chemical evolution, density then changes according to because they are a main source of iron-peak, r-ands-process elements. In the same way as for the SN II, we include a fidu- √ 8 π R2 cial chemical element to trace the enrichment with SN Ia nu- E˙ = −c · ρ2 σ3 . (12) cld coll 3 M cld cld cleosynthesis products. For the conversion into real abundances we use the chemical yield table of Nomoto et al. (1984). With the cloud mass-radius relation of Elmegreen (1989) we obtain 3.4.4. Intermediate mass stars ˙ = −η · −1/2 · ρ2 σ3 Ecld coll P4 cld cld (13) Intermediate mass stars act as a mass storage for the ISM. The mass loss of these stars is significant only at the end of the stel- with ηcoll = 0.025 for ccoll = 1 (geometrical cross section). lar evolution, when these stars enter the AGB and planetary Unfortunately, ηcoll is not well constrained, because magnetic nebulae phase. Intermediate mass stars end as white dwarfs fields, gravitational focusing, and self-gravity can influence the with masses between 0.5 M and 1 M. Unlike the massive effective cloud cross section significantly. ηcoll is the most un- stars, the intermediate mass stars eject only weakly enriched certain parameter in our ISM description. 968 M. Samland and O. E. Gerhard: Formation of a disk galaxy

3.4.7. Evaporation other hand the heating rate decreases, condensation and cloud formation set in. As a result, the hot gas density and therefore The different phases of the ISM exchange mass by evaporation the cooling efficiency drops and the gas temperature will stay as well as condensation and cloud formation processes. These nearly constant. processes depend mainly on the sizes of the cold clouds and the Some interactions are self-regulated by dynamical pro- density and temperature of the hot ambient gas. As in the two- cesses. For example, a higher star formation efficiency delays dimensional models (Samland et al. 1997), we use the model the settling of the clouds in the galactic disk, so that the av- of McKee & Begelman (1990) and Begelman & McKee (1990) erage density of the cloudy medium is lowered which in turn for the description of the evaporation, decreases the star formation rate again. For the moment we R / ρ˙ = 4.35 × 10−16 ρ · T 5 2. (14) neglect this (important) dynamical self-regulation, discussing evap M cld icm now only the influence of the important model parameters on 5 σ For 10 M clouds in the simple spherical cloud model of the velocity dispersion of the clouds cld, the hot gas temper- Sect. 3.3 we obtain an evaporation rate of ature Ticm, and the mass ratio between gas and clouds. We do this by calculating equilibrium conditions first numerically and ρ = η −1/4ρ · 5/2 ˙evap evapP4 cld Ticm (15) then in an analytical approximation. −19 The interaction network is a closed system of differential with ηevap = 1.0 × 10 .Thevalueofηevap depends on the cloud model (cloud shapes, mass spectrum). equations, which can be solved numerically to find equilibrium states for given densities of the ISM. The results are plotted in Fig. 4. As expected for a self-regulated system, we find that 3.4.8. Condensation and cloud formation even density variations by five orders of magnitude have only ff σ In small clouds, heat transfer by conduction is more effi- small e ects, changing cld (Ticm)byfactorsof3(7). cient than cooling by radiation (McKee & Begelman 1990). In For the analytical approach, we assume that, most of the large clouds (cloud radius exceeds the Field-length) the cool- ISM mass is in the cloudy medium and that the cloudy medium ing dominates and the clouds can gain mass by condensation. is in pressure equilibrium with the surrounding gas. Balancing The timescale for condensation is of the order of the cooling the heating by SNe with the dissipation by cloud-cloud colli- time of the ambient hot medium (McKee & Begelman 1990). sions and the cooling by radiation, and assuming that evapora- In addition, clouds can form when the hot gas becomes ther- tion rates are equal to condensation rates, we get the following mally unstable. This cloud formation process is important in three relations: high density regions (e.g., shocks) or in regions with no or low η η 4/5 σ ∝ sf cld · ρ−1/10 star formation activity. Since it is a cooling instability, this pro- cld η (17) cess also has a timescale which is approximately the cooling coll time. We use the following parameterization for the condensa- η η 2/7 η η 2/7 tion and cloud formation process: ∝ sf cld · sf cond · ρ1/7 Ticm η η (18) ρ coll evap ρ = η · icm · ˙cond cond (16) tcool ρ η η 3/10 η 1/2 icm ∝ sf cld · sf · ρ−7/20. It turns out that effectively η is not a free parameter. (19) cond ρcld ηcoll ηcool Reasonable values for ηcond are in the range of 1.0 (cloud formation time = cooling time) to 0.3. Tests showed that if Here ρ is the average density of the ISM (clouds and hot gas). ηcond < 0.3, the hot gas can cool to temperatures of less than These equations confirm that there is only a weak dependence 4 10 K before condensation and cloud formation are efficient. of σcld and Ticm on the average ISM density. They are shown in We use a value of ηcond = 0.5, which guarantees a stable hot Fig. 4 for comparison with the numerical results. gas phase. In the previous discussion, we neglected metallicity effects. However, metallicity can be important because it increases the cooling efficiency. The dashed lines in Fig. 4 show the nu- 3.4.9. Self-regulation merical equilibrium relations for an extremely metal-poor gas The interaction network describes a self-regulated system in ([Fe/H] = −4). The effects of low gas metallicities are similar which moderate changes of the parameters do not alter the re- to a low cooling efficiency ηcool. In both cases, the gas tem- sults significantly (see also Köppen et al. 1995). Star formation perature Ticm stays nearly constant, because the lower cooling is one example for such a self-regulation. An increase of the ef- efficiency is balanced by a higher gas density ρicm.Theself- ficiency ηsf by a certain factor will not increase the star forma- regulating character of the interaction network again is obvi- tion rate by the same amount, because the ISM will be heated ous. The equilibrium system is in a balance of forward and re- by the stars, which in turn decreases the amount of available verse reaction rates. Any stress that alters one of these rates cold gas to form stars. Another example is the phase transition makes the system shift, so that the two rates eventually equal- between hot and warm gas. A rise in the heating rate of the hot ize (Le Chatelier’s principle). gas leads to an enhanced evaporation of clouds. This increases In spite of the self-regulating character of the interaction the density of the hot gas and enhances the cooling. The tem- network it is necessary to solve the full interaction network dur- perature of the gas increases only by a small amount. If on the ing a simulation, because processes on different timescales are M. Samland and O. E. Gerhard: Formation of a disk galaxy 969

10.000 discussed in the subsections above. For ηevap and ηcoll,where [Fe/H]=-4 ρ-7/20 there is little constraint, we have used a large range to illustrate 1.000 that even in this case the resulting changes to the equilibrium cld ρ [Fe/H]=0 are not large. It is obvious from the analytical model above that / 0.100 η (η , η ) have inﬂuence on σ , T and ρ , while η

icm sf cld coll cld icm icm evap ρ (η ) can change only T , and by varying η one can shift 0.010 cond icm cool the gas-to-cloud mass ratio. Using the extreme values for the 0.001 parameters given in Table 1 we obtain the uncertainties of σcld, 0.001 0.010 0.100 1.000 Ticm,andρicm (Cols. 5–7 of Table 1). This shows that even large -3 ISM density [cm ] changes of the parameter values lead only to moderate variations in σcld, Ticm,andρicm. From these numbers we conclude that σcld, Ticm,andρicm/ρcld are accurate to a factor of 2 in our [Fe/H]=-4 ρ-1/10 model.

100 [Fe/H]=0

[km/s] 3.5. Gravitation cld σ

Gravitation is a long range force that couples the baryonic and 0.001 0.010 0.100 1.000 the dark matter and it is the main driver of the galactic evolu- ISM density [cm-3] tion. To determine the gravitational potential, we ﬁrst map all stars, the diﬀerent phases of the ISM and the dark matter onto one grid. Then, depending on the kind of grid (equidistant or logarithmically spaced) we solve the Poisson equation with a 7 ρ1/7 10 FFT (fast Fourier transformation) or a SOR (successive over- relaxation with Chebeyshev acceleration) method. The gravita-

[K] tional forces are calculated from the potential at every point.

icm [Fe/H]=0 T 106 [Fe/H]=-4 3.6. Initial conditions and code set-up 0.001 0.010 0.100 1.000 ISM density [cm-3] The present simulations used a logarithmic 813 grid with a spa- Fig. 4. The three panels show the density dependence of the intercloud tial resolution of 370 pc in the inner parts and 5 kpc in the outer σ to cloud mass ratio, the cloud velocity dispersion cld, and the hot gas halo. The timestep lengths, which in most cases are limited ff temperature Ticm in a closed-box model where dynamical e ects are by the interaction processes, are of the order of 104–105 yr. / = neglected. The dotted and dashed lines show the [Fe H] 0(solar The simulations start at a redshift z = 4.85 corresponding to metallicity) and the [Fe/H] = −4 cases and the full lines show the t = 1.2Gyrin a ΛCDM cosmology. Initially the dark halo slopes given by the analytical approximations of Eqs. (17)–(19). 10 has a mass of 2.1 × 10 M, distributed over a 30 kpc sphere. During the evolution, the halo grows by accretion to a size of 12 involved that compete with the dynamical evolution. For ex- 500 kpc with an enclosed mass of 1.8×10 M. The baryonic- ample, heating processes or cloud dissipation may take longer to-dark matter ratio of the accreted mass is fixed at 1:5, which 11 than dynamical changes. Equilibrium solutions of the interac- finally amounts to 3 × 10 M of baryonic matter inside the tion network can be used only to study the general behaviour r200 radius. The accretion rate, taken from cosmological simu- of a star-gas system where dynamical processes are slow, but lations (Kauffmann et al. 1999, see Sect. 3 and Fig. 5) decreases −1 −1 would not be able to describe, e.g., the outflow of hot gas. from 32 M yr at z = 4.85 (1.2Gyr)to8.1 M yr at z = 0 (13.5 Gyr). The baryonic matter outside the r200 radius consists of ionized (T > 104 K) primordial gas at the virial temperature. 3.4.10. Sensitivity to free parameters The cooling time of this gas is short compared to the collapse Most of the parameters in this multi-phase ISM description time. On this time-scale clouds form which can then dissipate (η’s) can be constrained either from theory or from observa- kinetic energy and collapse inside the dark halo. Thereafter the tions. The two most important parameters in the model are collapse is only delayed by stellar feedback processes. We have the star formation efficiency ηsf,andηcoll which controls the confirmed that starting with a cloudy gas medium leads to a dissipation rate. These parameters are of special interest be- very similar evolution. Initially, the accreted baryonic matter, cause they both influence the velocity dispersion of the cloudy like the dark matter, is in spherical rotation (Eq. (1)) with a medium and thus the settling of the baryonic matter. spin parameter λ = 0.05. It grows from inside-out and the mass In Table 1 we list the parameters of the interaction network with the highest specific angular momentum is accreted at late together with their upper and lower limit values. These are times. 970 M. Samland and O. E. Gerhard: Formation of a disk galaxy

Table 1. Parameters of the model interaction network and the equations in which they are deﬁned. Column 3 shows lower and upper limits for the parameters, Col. 4 the standard values used in the simulations, and Cols. 5–7 give the equilibrium σcld, Ticm,andρicm resulting with the maximum and minimum parameter values, normalized by those obtained for the standard model values.

σ / σstd / std ρ / ρstd parameter Eq. range std. value cld cld Ticm Ticm icm icm −4 −4 ηsf (8) (5–30) × 10 5.3 × 10 1.0–4.0 1.0–2.7 1.0–4.0 ηcld (10) 0.02–0.1 0.05 0.5–1.7 0.8–1.2 0.8–1.2 ηcool (11) 2.3–10 5 1.0 1.0 1.5–0.7 ηcoll (13) 0.013–0.5 0.025 1.7–0.1 1.2–0.4 1.2–0.4 −19 −19 ηevap (15) (0.1–10) × 10 10 1.0 1.9–0.5 1.0 ηcond (16) 0.3–1.0 0.5 1.0 0.9–1.2 1.0

43 2 1redshift 0 4.1. Collapse and star formation history 50 The accreted primordial gas quickly reaches an equilibrium between the cold and hot phases with most of the mass in the cold clouds. It can then dissipate its kinetic energy and begins to 40 collapse inside the dark halo. The strong collapse takes place in a redshift interval between z = 1.8(3.6 Gyr) and z = 0.8 (6.6 Gyr). With a duration of ∼3 Gyr, its period exceeds the r=r 30 200 free-fall time at the r200 radius by a factor of 3.4. This late and /yr]

o delayed collapse, which is in contradiction to simple collapse scenarios, is caused by stellar feedback in conjunction with the initially shallow gravitational potential of the halo.

MIR [M 20 The baryonic mass infall rate (MIR) across the r200 radius and into the inner 20 kpc is shown in Fig. 5. At early times the gas fallen through r200 has not yet reached the inner 20 kpc be- 10 cause of the delay from feedback. With the growing mass of baryons the dissipation increases and the infall into the centre accelerates, until the combination of feedback from the grow- r=r 20 kpc ing star formation rate and decreasing infall rate through r200 0 reverses this trend and the MIR into 20 kpc begins to decrease 0 2 4 6 8 10 12 14 z . z < . time [Gyr] again at 1 2. At late times ( ∼ 0 3) a significant fraction of the infalling gas has too much angular momentum to arrive in Fig. 5. Baryonic mass infall rate (MIR, full line) into inner 20 kpc and the central 20 kpc, so that the rate of infall into 20 kpc becomes across the r200 radius (dashed line). lower than that through r200. The mass of the galaxy increases 11 rapidly and at z = 1.2 (5 Gyr) already 10 M of baryonic matter has been accreted. At z = 0, 2/3 of the total baryonic 11 mass of 3 × 10 M has collapsed into the inner 20 kpc of the dark matter halo and has formed the luminous galaxy, leading 4. The formation history to a dark-to-baryonic matter ratio of 3%, 10% and 30% inside The present simulation describes the formation and subsequent galactocentric radii of 1, 3, and 10 kpc, respectively. evolution of a disk galaxy in a slowly growing dark halo. The The star formation rate integrated over the central 20 kpc halo grows by accreting dark and baryonic matter in spherical is shown in Fig. 6. This increases steeply to a maximum of ≈ −1 = . shells with an accretion rate that is derived from cosmologi- 50 M yr at z 1(575 Gyr). At that time, the gas con- cal simulations in a ΛCDM universe (Kauffmann et al. 1999, sumption by the star formation process is balanced to 80% by see Sect. 3.1), with a baryonic mass fraction of 1/6. The dark infall and to 20% by the stellar mass return. Afterwards the in- −1 = and baryonic matter have initial λ = 0.05 and specific angu- fall rate decreases to 5 M yr ,atz 0, while the stellar mass −1 lar momentum distribution similar to that from Bullock et al. return stays nearly constant at a rate of 10 M yr . This ex- ff (2001). Star formation, feedback, and gas physics of the bary- plains the di erent shapes of the curves in Figs. 5 and 6 at late onic component are described within a two-phase ISM model times. (see Sect. 3.3). The dynamics of the gas and stars is treated = . = . separately. The simulations start at z 4 85 (t 1 2Gyr) 4.2. Sequential formation of halo, bulge and disk with a mixture of dark matter and gas, but no stars. At red- 10 shift z = 4.85 the dark halo mass exceeds 2.0 × 10 M,which Figures 7, 8 show the spatial distributions of the gas and stars is twice the mass resolution of the cosmological simulations on as a function of redshift. In general terms, star formation out of which our halo model is based. the dissipating cloud medium proceeds from halo to disk and M. Samland and O. E. Gerhard: Formation of a disk galaxy 971

43 2 1redshift 0 (ii) higher gas density and, thus, dissipation rate for gas near the angular momentum barrier in a deeper potential well. The 50 ﬁnal disk thickness in Fig. 10 of ≈500 pc is set by the resolution the present model. After the bar-bulge has formed, two trailing spiral arms ap- 40 pear in the disk which are connected to the bar-bulge. At the beginning the two spiral arms are symmetric and of the same size. However, in the further evolution a persistent lopsidedness of the galaxy develops. The resulting oﬀ-centre motion of /yr]

o 30 the bulge-bar brings one arm nearer to the bulge-bar, and the more distant spiral arm becomes more prominent over a wider

SFR [M winding angle (Fig. 7). The lopsidedness may be produced by 20 the dark halo which is here described as an external spherically symmetric potential component, even though in the inner galaxy the baryonic matter dominates the gravitational poten- 10 tial. The effect that such perturbations can produce lopsided galaxies is well known (Levine & Sparke 1998; Swaters et al. 1999). 0 We expect this evolutionary sequence to be more or less 0 2 4 6 8 10 12 14 typical of any dissipative collapse in a growing dark matter time [Gyr] halo, unless it is interrupted by substantial mergers. The follow- Fig. 6. Star formation rate (SFR) in the inner 20 kpc of the dark halo. ing modelling uncertainties may further modify the evolution The star formation rate peaks slightly later than the MIR (Fig. 5). as described above. (i) The assumed dissipation parameter is uncertain. If in reality dissipation is significantly more efficient than in the model, and in addition stars form only above a certain density threshold (Kennicutt 1998), the gas would fall into from inside out. Thus the earliest stars at redshifts z > 2formin the disk much more rapidly without significant star formation. In this case formation of stars in the halo and thick disk may be the whole volume limited by the r200 radius (halo formation), with a concentration to the central bulge region. At a redshift of substantially suppressed. Also, in this case the velocity disper- z = 1.3(4.7 Gyr), a (thick) disk component first appears. Later sion of the cloudy medium in the disk would decrease. Then the star formation is concentrated to the equatorial plane, and the gaseous disk may fragment in local instabilities before a at late times most of the star formation occurs in the outer disk. global disk instability sets in, so that more material would dissipate to the centre and contribute to the bulge. (ii) The pre- Around z = 1.2 the infall of gas can no longer compen- cise angular momentum distribution (as compared to the uni- sate the gas consumption by star formation in the inner disk. versal average distribution assumed) will influence the detailed This is because the infalling molecular cloud medium now has inside-out formation of the disk. For example, the formation of higher specific angular momentum, so that its infall timescale a ring in the present model would likely be suppressed by in- becomes longer than the central star formation timescale. As a creasing the amount of low angular momentum gas falling in result, the region of highest star formation density moves radi- at redshift z 1. (iii) When small-scale initial fluctuations lead ally outwards from the centre and a ring (Fig. 7, z = 1 panels) to the formation of sub-galactic fragments that merge before forms. This ring grows to a radius of ∼4 kpc before, at redshift the main collapse, the mass of the bulge component would be z = 0.85 (6.4 Gyr), the disk becomes unstable. Within 150 Myr substantially greater, and its dynamical structure determined by the ring then fragments, and for a short time (≈150 Myr) a the merging process (Steinmetz & Navarro 2002). very elongated bar is formed with exponential major-axis scale- length of 5.9 kpc and axis ratio of 12:2:1. Figure 9 shows that after this transient period the bar length decreases, and a bar- 4.3. Global chemical evolution bulge with axial ratios ∼3:1.4:1 develops which includes the old bulge component formed in the early collapse. The chemical evolution of the model proceeds in three steps. Already during this bar formation process the galaxy starts First, there is the collapse phase, lasting until z 0.6. Then two to build up the outer disk. This disk is the youngest component quasi-equilibrium phases follow in which first (z = 0.6−0.3) in the model galaxy, even though the oldest disk stars are as the mass infall and later (z = 0.3−0.0) the stellar mass return old as the halo stars. The disk grows from inside-out, because determines the star formation rate (see Figs. 5 and 6). These the early accreted mass has low specific angular momentum three phases are clearly visible in the chemical enrichment his- (see the last columns of Figs. 7 and 8). In parallel, the vertical tory of the cloudy medium. This is shown in Fig. 11, where we scale-height in a fixed radial range decreases; this is shown in have plotted the Zn/H metallicity normalized to the solar value Fig. 10 for the range 3−10 kpc. The more pronounced settling because Zn is believed to not be depleted onto dust grains. The of the cloudy medium to the equatorial plane at late times is Zn/H abundance was reconstructed from the fiducial SN I and due to (i) more efficient cooling because of higher metallicity, SN II elements as described in Sect. 3. 972 M. Samland and O. E. Gerhard: Formation of a disk galaxy

Fig. 7. Face-on surface density of the ionized gas, the cloudy medium, the OB-associations and the stars at diﬀerent redshifts. Each column shows the evolution of one component between redshift z = 2andz = 0. Each panel has a size of 50 × 50 kpc. M. Samland and O. E. Gerhard: Formation of a disk galaxy 973

Fig. 8. Same as Fig. 7, but for the edge-on view. 974 M. Samland and O. E. Gerhard: Formation of a disk galaxy

43 2 1redshift 0

1.0 stars

0.8 molecular gas

0.6 Zn/H

0.4 ionized gas

0.2

0.0 0 2 4 6 8 10 12 14 time [Gyr]

Fig. 11. Average Zn/H abundance (linear scale) of the ISM and the stars in the inner 20 kpc of the model galaxy. Fig. 9. Evolution of the bar-bulge axis ratios. z indicates the vertical direction and x and y the major and minor axes of the bar. The bar . forms from a disk instability at time 6 4 Gyr and settles to a triaxial The chemical enrichment histories of the hot gas, cloudy ∼ bulge with axis ratios 3:1.4:1 at late times. medium, and stars (shown separately in Fig. 11) are very different from the predictions of multi-phase closed box models. In these models the hot ionized gas always has the highest metallicity, followed by the molecular cloud medium and the stars. This is different in our model, because stars, clouds and gas have different spatial distributions. Most of the ionized hot gas is located in the halo, while the molecular clouds and stars are preferentially found in the disk and bulge, where the gas densities and metallicities are high. After a brief initial period the average metallicity of the stars exceeds that of the cloudy medium, because the stars form from gas that is more concentrated to the equatorial plane and thus more metal-rich than the average cold gas medium. An interesting feature in Fig. 11 is the constant [Zn/H] ≈ 0.4 of the hot gas from redshift z = 1 until the present epoch. A large fraction of this gas occupies the galactic halo, Fig. 10. Evolution of the vertical scale height of the stellar disk be- where the star formation rate is low. Two processes keep the tween radial distances 3−10 kpc. This part of the disk forms later than halo metallicity at a constant level: Gas flows from the bulge the average of all stars shown in Fig. 5. The scale heights in small and the disk transport heavy elements into the halo. There it rings are shown as dots and the radial average is drawn as a full line. mixes with the existing ISM and with infalling low metallic- The dashed line indicates the spatial resolution limit of 370 pc. ity gas. In this way the metallicity in the halo gas can remain constant over a long time. A similar enrichment can be observed in the disk. During The initial collapse of the baryonic material produces a the collapse the star formation and the metal production is con- first generation of stars which already at z = 2 has synthe- centrated to the inner galaxy. From there, the metal-rich gas sized enough metals to increase the average metallicity to ≈−1. expands into the disk. This disk pre-enrichment causes a lack During the late collapse phase the star formation rate exceeds of metal-poor stars (“G-dwarf problem”1, see Sect. 5), similar the MIR and the metallicity increases fast. Then in the second to that observed in the Milky Way. phase, the star formation rate is proportional to the MIR and the metallicity increases only slowly. In the final phase, the star 1 The term “G-dwarf problem” describes the fact that the fraction formation rate is again higher than the MIR and the metallicity of low-metallicity stars in the Milky Way disk is substantially lower increases more rapidly (Fig. 11). than predicted by simple 1-zone models. M. Samland and O. E. Gerhard: Formation of a disk galaxy 975

We find a global age-metallicity relation and also a [O/Fe] 54 3redshift 2 to [Fe/H] relation, which are consistent with Milky Way ob- 1 servations. These relations are much more influenced by the choice of stellar yields, resp. the star formation history, than by the details of the galactic model, and have been found in 0 previous galactic models as well (e.g. Steinmetz & Müller 1994; Timmes et al. 1995; Samland 1998). However, the pre-enrichment of the galactic disk, explaining the so-called “G-dwarf problem” and the sub-solar abundances in the outer -1 halo gas are a result of dynamical processes and are therefore strongly influenced by the galactic model. The same is true for relations connecting the kinematics and chemical composition [Zn/H] of stars; see Sect. 5.7. -2 In Fig. 12 we compare our results with the [Zn/H] data for damped Lyα systems (Pettini et al. 1999; Prochaska et al. / / 2001), using [Zn H] because [Fe H] may be depleted by dust. -3 Damped Lyα systems are an important test for galactic models. If they are associated with forming disk galaxies, they can be used to measure the metallicity of the ISM of young, gas rich galaxies at high redshifts. The Lyα systems on average show -4 only a mild evolution in [Zn/H] for z ≈ 2−3. Even though we 1 2 3 4 time [Gyr] consider only a single model with its specific evolutionary history, the predicted metallicity range as a function of redshift Fig. 12. Redshift evolution of the average zinc abundance [Zn/H] of is consistent with the data. From the model we thus infer that the cloudy medium (full line) inside a galactocentric radius of 20 kpc. the time needed to increase the disk metallicity to the typical The shaded area shows the range of [Zn/H] metallicities in the model values observed in Lyα systems is ∼1 Gyr. The shaded area in over this region. The points are data for damped Lyα systems from Fig. 12 shows the range of metallicities from different loca- Pettini et al. (1999) and Prochaska et al. (2001), plotted as squares tions in the model at redshifts z > 1.6. Figure 12 also shows the and rhombi, respectively. The general trend of the model curve is consistent with the observed damped Lyα abundances. The dashed line early [Zn/H] evolution of the hot ionized gas (dashed line). This indicates the metallicity evolution of the hot gas component, which should be representative for the average metallicity of metal- fills most of the galactic halo. enriched clouds forming in the low density halo ISM, and thus for absorption of background quasars along halo lines-of-sight. spatially resolved dynamical and chemical information for all of these components. With this information it is possible to 4.4. The ISM at z = 0 study the detailed consequences for the stellar kinematics and 10 At z = 0, 5.6 × 10 M (15.7%) of the initial baryonic mass of abundances, that are implied by the assumptions made for the 11 3 × 10 M is still in the form of gas. The distribution of this galaxy formation scenario and the modelling of the various late ISM is spatially more extended than that of the stars and physical processes involved, and to compare to a variety of ob- the mass ratio of ISM to stars increases with distance from the servational data. galactic centre. Inside 10 kpc, only about 2% of the baryonic Figure 13 shows average properties of the model stars at mass is still in the ISM, but inside 60 kpc the fraction is already redshift z = 0: age, metallicity, [O/Fe], and rotational velocity. 43%. 75% of the ISM mass is in the cold and warm phases These vary continuously as a function of position in the merid- (molecular and atomic gas), but a significant fraction of 25% is ional plane (R, Z), in a way that reflects the interplay between hot and ionized. This hot gas extends far out into the halo and star formation and enrichment on the one hand, and the verti- fills most of the halo volume (see Fig. 8). As discussed in the cally top-down and radially inside-out dynamical formation on previous subsection, the hot gas is the chemically least enriched the other hand. Thus, the oldest stars can be found at low radial component at late times. distances in the halo and the youngest stellar populations are located in the outer galactic disk. Because mean age and mean metallicity vary as a function of position, global mean values 5. Formation of galactic components for age or metallicity are not sufficient to characterize the stel- The model presented in this paper describes the formation of a lar population of a galaxy. Notice also that stars at a given mean 11 massive disk galaxy with a total luminous mass of 2 × 10 M age or metallicity can be found in a variety of positions. inside 20 kpc and a disk rotation velocity of 270 km s−1 at a In Fig. 14 we plot the complete distribution of model stars radius 10 kpc, about 2.5 times more massive than the Milky at redshift z = 0 in the plane of rotational velocity vrot ver- Way. The galaxy model forms from inside-out radially and sus metallicity [Fe/H]. This shows the expected evolution from from halo to disk vertically. In Fig. 6 one can clearly see halo, non-rotating, low-metallicity stars formed early in the collapse bulge, and disk components, which as described in Sect. 4 to a rotating disk-like, metal-rich population formed at late form sequentially in this order. The model contains the full times. However, this evolution is by no means described by a 976 M. Samland and O. E. Gerhard: Formation of a disk galaxy

9 7 7 8 8 10 stellar age [Fe/H]

7 6 -1.50 -1.40 25

20 8 -1.30

7 -1.40 15 9 8

z [kpc] 6 10 -1.10 10 -1.20

5 -1.00 -0.90 5 -0.80 -0.70-0.60 4 -0.40-0.50 -0.30 0 -0.20 25 [O/Fe] 50 rotation velocity 50 25 0 0.325 0.300 75 20 0

0.275 25

50 0.250 100

15 75 z [kpc]

75 10 0.225

0.200 5 0.175 0.150 175 100 150 225 0.125 125 200 0 0.100 250 0 5 10 15 20 25 0 5 10 15 20 25 r [kpc] r [kpc]

Fig. 13. The panels show the mean stellar age, [Fe/H], [O/Fe], and mean rotation velocity V of the model stars at redshift z = 0, in the meridional plane as a function of radial (R) and vertical distance (Z) from the galaxy’s centre. The mean stellar ages are given in Gyr, and the units of the velocities are pc Myr−1 = 0.98 km s−1.

single dependence of vrot on [Fe/H]; rather there is large scat- and baryonic masses of this model are larger than in the Milky ter in vrot at given metallicity, especially at low [Fe/H] values. Way, and especially the dissipation parameter of the model may In addition, there are also at least two concentrations of stars not be typical for the Milky Way. Nonetheless, studying these visible, at [Fe/H] −0.4and[Fe/H] 0.1. components is useful to understand the course of the dynamical The top panel of Fig. 14 shows a [Fe/H] histogram of all evolution and chemical enrichment in dissipative, star-forming stars, projecting along the vrot axis. Based on this histogram, collapse, and we will for definiteness use the Milky Way ter- we will now study the stellar populations in the different parts minology in the following discussion of the properties of these of Fig. 14 separated by the dashed lines. In first order these populations. might be described as extreme halo, inner halo, metal-weak thick disk, thick disk, thin disk, and central bulge components, 5.1. [Fe/H] < −1.9 (“Extreme halo”) somewhat reminiscent of the similar components known in the Milky Way. However, we do not intend to imply a detailed cor- This component comprises all stars with [Fe/H] < −1.9; these respondence with the Milky Way components; there are signif- stars form before the collapse and simultaneous spin-up of the icant differences with the present model. Recall that the dark gas goes under way. As the left row of Fig. 15 shows, these stars M. Samland and O. E. Gerhard: Formation of a disk galaxy 977

[Fe/H] distribution: all stars outﬂowing enriched material. This younger component may not form in real galaxies if stars can only form above a density threshold, as inferred from observations by Kennicutt (1998). Further simulations also showed that a larger dissipation parameter (see Sect. 3.4.6) leads to a faster settling of the clouds into the disk and this suppresses the star formation in the halo at late times.

500

400 5.3. −0.85 < [Fe/H] < −0.6 (“Metal-weak thick disk”) 300 These stars with −0.85 < [Fe/H] < −0.6 are intermediate in 200 their rotational properties between a halo and a disk population (see Fig. 14). As Fig. 15 shows their spatial distribution

[pc/Myr] 100

rot is already strongly ﬂattened. They have a sizeable rotation with

v −1 0 peak of the distribution at vrot 150 km s , around 50–60% of the circular velocity. The eccentricity distribution is broad and -100 centred around 0.5. The oldest stars of this component form at -200 around the peak of the global star formation rate when the gas -3 -2 -1 0 still has [O/Fe] 0.3. A slightly larger fraction of stars in this [Fe/H] range of [Fe/H] forms continuously until late times, at progres- Fig. 14. Lower panel: distribution of model stars in the metallicity– sively larger radii, again favoured by the lack of star formation rotation velocity plane. Rotation velocities are perpendicular to the threshold. These stars are already enriched by SN Ia, as visible total angular momentum vector and are in pc Myr−1. Upper panel: in their [O/Fe] distribution. total metallicity distribution of all model stars, projecting along the rotation velocity axis. The dashed lines separate the subpopulations discussed in the text. 5.4. −0.6 < [Fe/H] < −0.15 (“Thick disk”) This component with −0.6 < [Fe/H] < −0.15 is one of the dis- form mostly in the first two Gyr after the start of the simulation tinct components in Fig. 14, and is clearly dominated by a disk (at time 1.2 Gyr), but there is a small ongoing star formation component in its rotational properties and eccentricity distribu- in the halo even to late times from infalling low-metallicity gas tion. However, Fig. 14 shows that it also contains intermediate (remember that we have not imposed a density threshold for metallicity retrograde stars, showing that this metallicity selec- star formation in this model). These stars have [O/Fe] near 0.4 tion also includes part of the bulge. These bulge stars cause the ∼ / typical for enrichment by SNe II. Their spatial distribution is peak in the age distribution at 5 Gyr, with high [O Fe]. The / nearly spherical, they rotate slowly if at all, and they have an disk part has a relatively narrow distribution in [O Fe] centred ∼ . asymmetric, broad distribution of eccentricities biased towards around 0 13. radial orbits. Here we have defined eccentricity as e = (rmax − / + rmin) (rmax rmin)wherermax and rmin are the maximum and 5.5. −0.15 < [Fe/H] < 0.17 (“Thin disk”) minimal three-dimensional radii of the orbit. The relative lack of low-eccentricity orbits at these metallicities (see Sect. 5.7) The thin disk component, selected in the interval −0.15 < may be due to our neglect of small-scale structure in the initial [Fe/H] < 0.17, is characterized by near-circular orbits and a conditions; compare with Bekki & Chiba (2001). high rotation velocity, about 270 km s−1. This component forms with an approximately constant star formation rate and has mostly near-solar [O/Fe]. As in the previous thick disk compo- 5.2. −1.9 < [Fe/H] < −0.85 (“Inner halo”) nent, a contribution from bulge stars is included by the metal- Stars in this component have a metallicity range −1.9 < licity selection. The thin disk shows the bar and spiral arms [Fe/H] < −0.85, covering the spin-up phase until the first min- clearest; see the top panels in Fig. 15. In time, it forms from imum in the [Fe/H] histogram in Fig. 14. These stars form a inside out due to the accretion of material with progressively more flattened spheroidal distribution with a flat density pro- larger angular momentum. file. They rotate with ∼70 km s−1, and the eccentricity distribu- The thin disk shows only a shallow radial metallicity gradi- tion is less radially biased than for the extreme halo compo- ent of 0.02 dex/kpc, which is due to efficient mixing of the disk nent. This component forms partly during the phase of rapid ISM by the galactic bar. Beyond 16 kpc, where the mixing is increase in the global star formation rate (Fig. 6), but there is not very efficient, a steeper metallicity gradient of 0.07 dex/kpc also a younger part forming until late times. Here the selection is established. See also Martinet & Friedli (1997) who found in metallicity is important: the older part of these stars forms strong metallicity gradients only in non-barred galaxies. The at relatively small radii when the metallicity there reaches the metal production by the bulge and the mixing by the bar lead selected range, while the younger component forms at progres- to a pronounced pre-enrichment of the disk, which is visible in sively larger radii from infalling low-density gas mixed with Fig. 16 by the lack of metal-poor stars. 978 M. Samland and O. E. Gerhard: Formation of a disk galaxy

Fig. 15. Properties of model subcomponent selected by metallicity according to the dashed lines in Fig. 14. Columns from left to right show all stars with [Fe/H] < −1.9 (“Extreme halo”), −1.9 < [Fe/H] < −0.85 (“Inner halo”), −0.85 < [Fe/H] < −0.6 (“Metal-weak thick disk”), −0.6 < [Fe/H] < −0.15 (“Thick disk”), −0.15 < [Fe/H] < 0.17 (“Thin disk”), and [Fe/H] > 0.17 (“Inner bulge”). For each of these components, the panels from top to bottom show the face-on projection onto the disk plane, edge-on projection, distribution of formation times, [O/Fe] distribution, eccentricity distribution, and distribution of rotation velocities. The top two panels in each row show an area of 40 kpc × 40 kpc. Star formation in the model starts at time 1.2Gyr. M. Samland and O. E. Gerhard: Formation of a disk galaxy 979

number of stars

closed box model -1.5 -1.0 -0.5 0.0 [O/H]

Fig. 16. Distribution function of oxygen abundances of G-dwarfs, selected by mass, in the model galaxy at redshift z = 0. All G-dwarfs in the equatorial plane with galactocentric distances between 9.5 kpc and 10.5 kpc are plotted. The dotted curve shows predictions of an instantaneous simple model with an eﬀective yield Yeﬀ = Z.

age distribution is similar to that of the thick disk component. As the lower panels of Fig. 15 show, the metal rich inner bulge forms from z 1 in this model, rotates relatively slowly and has a broad eccentricity distribution. The total bulge population includes this inner component and a second, more metal-poor component included in the metallicity range of the “thick disk”. Two subpopulations in the bulge is consistent with the observations of Prugniel et al. (2001), who ﬁnd that galactic bulges in general consist of two diﬀerent stellar components, an early collapse population and a population that formed later out of accreted disk mass.

5.7. Kinematics-abundance relations The previous subsections have shown that while it is useful to select stellar components by metallicity, such a selection is not ideal because it lacks spatial and kinematical information. Vice versa, a purely kinematical selection would mix stars from halo and bulge, or metal-poor and metal-rich disk. A more physical separation would involve spatial, kinematic, and abundance information. This is possible with models such as Fig. 15. continued. this, because the complete population of model stars is available, but will be deferred to a later paper. Here, we will only show two relations between kinematic and abundance parame- 5.6. [Fe/H] > 0.17 (“Inner bulge”) ters for all stars in the model, which further illustrate some of its properties. The ﬁnal metallicity component with [Fe/H] > 0.17 consists Figure 17 shows the mean stellar rotation velocity as a func- mainly of stars in the central bulge. Notice that these most tion of metallicity [Fe/H] for all model stars. For [Fe/H] < −1 metal-rich stars do not form preferentially at late times; their −2 vrot is independent of metallicity, at about 50 km s .Itthen 980 M. Samland and O. E. Gerhard: Formation of a disk galaxy

300 -3.0 250

200 -2.5

150 -2.0 100

50 -1.5 [Fe/H]

rotation velocity [pc/Myr] 0 -50 -1.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 [Fe/H] -0.5 Fig. 17. Mean stellar rotation velocity as a function of [Fe/H] for all model stars. 0.0 0.0 0.2 0.4 0.6 0.8 1.0 eccentricity rises roughly linearly with [Fe/H] until the disk rotation veloc- / . Fig. 18. Sample of stellar orbital eccentricities and mean eccentricity ity is reached at [Fe H] 0 4. As is apparent from Fig. 14, of all stars in the model, as a function of metallicity. there is a large dispersion around the mean value except in the disk components. Figure 18 shows the mean orbital eccentricity and a sample of individual stellar eccentricities (defined in 6. Conclusions Sect. 5.1) as a function of metallicity. It is clear that at any metallicity, there is a large scatter in the orbital shapes. One of We have presented a model for the dissipative formation of a the causes of this is the very substantial deepening of the gravi- large disk galaxy, inside a growing dark matter halo according tational potential during the prolonged infall, which also affect to the currently favoured ΛCDM cosmology with ΩΛ = 0.7, ff orbits of stars formed early in the collapse. A second e ect is Ω0 = 0.3, h0 = 0.7. The dark halo mass accretion history is that the turbulent velocities in the cloud medium are substantial taken from the large-scale structure formation simulations of during the collapse and during periods of large star formation the GIF-VIRGO consortium (Kauffmann et al. 1999). We use a 12 rate, so that the stars made from these clouds inherit significant total dark matter mass of 1.5×10 M, a total baryonic mass of 11 initial velocities. Thus all eccentricity distributions shown in 3×10 M, a spin parameter λ = 0.05, and an angular momen- Fig. 15 are fairly broad. When the various distributions shown tum profile similar to the universal profile found by Bullock in that figure are added with the appropriate mass weighting, a et al. (2001). broad distribution results. We model the dynamics of stars and of two phases of Both the dependence of vrot on metallicity in Fig. 17 and the interstellar medium, including a phenomenological descrip- that of eccentricity on [Fe/H] in Fig. 18 are similar to those ob- tion of star formation, the most important interaction processes served in the Milky Way near the Sun (Chiba & Beers 2000), between stars and the ISM, and the feedback from massive even though our model was not made to match the Milky Way stars and SNe. The dynamical evolution of the stars is treated and does not in detail describe the Milky Way. The main con- with a particle-mesh code. For the dynamical description of clusion we draw from diagrams like Figs. 17 and 18 is that the cold molecular and hot ionized gas components we use realistic dissipative collapses are much more complicated than a three-dimensional hydrodynamical grid code. Linked to the the simple model put forward by Eggen et al. (1962) forty years N-body and hydrodynamical code is a Poisson solver for the ago. Therefore, inferring a merger history from the lack of ro- self-gravity and an interaction network, including a description tation gradient with metallicity and a broad eccentricity distri- of star formation, the heating and enrichment of type Ia and II bution in the Milky Way halo is premature, irrespective of the SNe, mass return by intermediate mass stars, the heating and fact that a fraction of the halo is likely to have been formed cooling by radiation, dissipation in the cloudy medium, for- out of merging fragments (Ibata et al. 2001; Helmi et al. 1999; mation and evaporation of cold clouds in the hot intercloud Chiba & Beers 2000). The important question, about the rel- medium. We discuss the range of uncertainty in the parame- ative importance of the disruption of small scale stellar frag- ters that enter this description. Because of the self-regulation of ments, merging of gaseous fragments, and star formation in these processes that is quickly established, even large changes smooth dissipative accretion, all within their dark matter halos, in these parameters only lead to moderate changes, about a fac- will need more detailed modelling and more data to answer. tor of 2, in the cloud velocity dispersion, hot gas temperature All of these processes are expected in current CDM galaxy for- and ratio of hot to cold gas density. mation theories. The present simulations show that feedback is The simulation starts at a redshift of z = 4.85 with a dark 10 important for the final properties of the stars formed. halo of 2.1 × 10 M. Inside the growing dark halo a disk M. Samland and O. E. Gerhard: Formation of a disk galaxy 981 galaxy forms, whose star formation rate reaches a maximum 8. The most metal-rich stars form approximately 1 Gyr after −1 of 50 M yr at redshift z ≈ 1. Our main results can be sum- the peak of the star formation, when the SN Ia rate is at its marized as follows: maximum. These stars have the lowest [α/Fe] ratios and, on 1. The formation and evolution of a galaxy in a dissipa- average, an age of ∼5 Gyr in this model. The infall of gas and tive collapse scenario is much more complex than predicted by the mass return from old stars decreases the average metallicity simple models. The energy release by SNe and massive stars in the inner galaxy at later times. (feedback) prevents the protogalaxy from a rapid collapse and This chemo-dynamical model provides kinematics and delays the peak in the star formation to a redshift of z ≈ 1. The metallicities of individual stars, but also can be used to ob- prolonged formation time causes fundamental changes in the tain colours, metallicities and velocity dispersions of integrated galactic shape, the kinematics of the stars, and the distribution stellar populations. It can therefore be used as a tool to under- of the heavy elements. stand observations of distant spiral galaxies, and connect them 2. The galaxy forms radially from inside-out and vertically with stellar data in the Milky Way. In addition, it provides gas from halo-to-disk. The dynamics of the collapse strongly in- phase metallicities and temperatures, and star formation rates, fluences the star formation and chemical enrichment history, as a function of time or redshift. Westera et al. (2002) have al- by modifying the gas density, cooling time, and thus star for- ready used this information to predict the colour evolution of mation rate. The on average youngest and oldest stellar pop- large spiral galaxies, and have compared with bulge colours in ulations are found in the outer disk and in the halo near the the Hubble Deep Field. Further “observing” of such models galactic rotation axis, respectively. As a function of metallicity, and comparing with distant galaxy observations may lead to we have described a sequence of populations, reminiscent of valuable insights on the galaxy formation process. the extreme halo, inner halo, metal-poor thick disk, thick disk, A table of the positions, velocities, and metallicities for all thin disk and inner bulge in the Milky Way. The first galactic stars in the model can be obtained by request from the authors. component that forms is the halo, followed by the bulge and Acknowledgements. Most of this work was supported by the Swiss the disk-halo transition region, and as the last component the Nationalfond. We thank M. Pettini for sending us the updated list disk forms. of Zn abundances in Lyα systems, and M. Steinmetz for a careful 3. An interesting feature of this model is the formation of a referee report. 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